Show that

For any real number x, let {x} = x −[x] denote the fractional part where [x] is the
usual floor function.
(a) Show that there exist infinitely many positive rationals x such that {x}+{x^2} = 0.99.
(b) Show that there are no positive rational x such that {x} + {x^2} = 1.

There are infinitely many non-negative integers and hence infinitely many such rationals

(b)

Note that for any Hence, in order to have we must have If is a positive non-integer rational, would have more decimal places than This would mean that their fractional parts cannot add up to 1. Hence there are no positive rationals such that

Remark: There do however exist positive irrational numbers with To see this, note that is continuous on the interval and while the result follows by the intermediate-value theorem.